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A254526
Fourier expansion of first basis element of space of weight 1 modular forms on Gamma1(12).
1
1, 0, 0, 0, 0, 12, 0, -12, 0, 0, 12, 0, 0, 0, -12, 12, 0, 12, 0, -12, 12, -12, 0, 0, 0, 12, 0, 0, -12, 12, 12, -12, 0, 0, 12, 0, 0, 0, -12, 0, 12, 12, -12, -12, 0, 12, 0, 0, 0, -12, 12, 12, 0, 12, 0, 0, -12, -12, 12, 0, 12, 0, -12, -12, 0, 24, 0, -12, 12, 0
OFFSET
0,6
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of 3 * (phi(q)^2 + phi(q^3)^2) / 2 - (a(q) + a(q^2)) in powers of q where phi() is a Ramanujan theta function and a() is a cubic AGM theta function.
Moebius transform is period 12 sequence [ 0, 0, 0, 0, 12, 0, -12, 0, 0, 0, 0, 0, ...]. - Michael Somos, Jan 31 2015
EXAMPLE
G.f. = 1 + 12*q^5 - 12*q^7 + 12*q^10 - 12*q^14 + 12*q^15 + 12*q^17 - 12*q^19 + ...
MATHEMATICA
a[ n_] := If[ n < 1, Boole[n==0], 12 Sum[ Boole[ Mod[d, 12] == 5] - Boole[ Mod[d, 12] == 7], {d, Divisors @ n}]];
PROG
(PARI) {a(n) = if( n<1, n==0, 12 * sumdiv( n, d, (d%12 == 5) - (d%12 == 7)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 3 * eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^6 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^12 + A)) - 2 * eta(x^2 + A)^6 * eta(x^3 + A) / (eta(x + A)^3 * eta(x^6 + A)^2), n))};
(Magma) Basis( ModularForms( Gamma1(12), 1), 70)[1];
CROSSREFS
Sequence in context: A370526 A048730 A307163 * A156390 A059680 A307170
KEYWORD
sign
AUTHOR
Michael Somos, Jan 31 2015
STATUS
approved